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Chapter 4: Problem 360

Use slopes and \(y\) -intercepts to determine if the lines are parallel. \(x-y=2 ; \quad 2 x-2 y=4\)

Short Answer

Expert verified
The lines are parallel because they have the same slope of 1.

Step by step solution

01

Convert the First Equation to Slope-Intercept Form

The slope-intercept form of a line is given by ewline \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Starting with the equation \( x - y = 2 \), solve for \( y \): Subtract \( x \) from both sides: \(-y = -x + 2\) Multiply both sides by \( -1 \): \( y = x - 2 \) The slope \( m_1 \) of the first line is 1.
02

Convert the Second Equation to Slope-Intercept Form

Starting with the equation \( 2x - 2y = 4 \), solve for \( y \): Divide both sides by 2 to simplify: \( x - y = 2 \) We have already converted this equation in Step 1, which resulted in \( y = x - 2 \). The slope \( m_2 \) of the second line is also 1.
03

Compare the Slopes

Since both slopes are equal (\( m_1 = 1 \) and \( m_2 = 1 \)), the lines are parallel. Parallel lines have identical slopes.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

slope-intercept form
The slope-intercept form is a way of writing the equation of a line so that you can easily see the slope and the y-intercept. The general form is \[ y = mx + b \] where \( m \) is the slope and \( b \) is the y-intercept. Knowing the equation in this form makes it straightforward to graph the line or to compare it with other lines. For example, if the equation of a line is \( y = 2x + 3 \), then the slope \( m \) is 2 and the y-intercept \( b \) is 3.
slope
The slope of a line measures its steepness or tilt. In the slope-intercept form \( y = mx + b \), the slope is represented by \( m \). It tells you how much the y-value of the line changes for a change in the x-value. If the slope is 1, like in our exercise \( y = x - 2 \), it means that for every unit increase in x, the y-value increases by 1 unit.

Simply put:
  • A positive slope means the line goes up as you move from left to right
  • A negative slope means the line goes down as you move from left to right
  • A zero slope means the line is horizontal
  • An undefined slope means the line is vertical
y-intercept
The y-intercept is where the line crosses the y-axis. In the equation \( y = mx + b \), the y-intercept is represented by \( b \). For instance, in the equation from our step-by-step problem, \( y = x - 2 \), the y-intercept is -2. This means that when \( x = 0 \), \( y = -2 \). It's the starting point of the line if you begin at x = 0.
linear equations
Linear equations describe linear relationships between x and y. They can be graphed as straight lines and often come in forms like:
  • Standard Form: \( Ax + By = C \)
  • Slope-Intercept Form: \( y = mx + b \)
  • Point-Slope Form: \( y - y1 = m(x - x1) \)
For the equations \( x - y = 2 \) and \( 2x - 2y = 4 \) from the exercise, we convert them to slope-intercept form to make them easier to graph and understand.
algebra
Algebra is the branch of mathematics that deals with symbols and the rules for manipulating those symbols. In solving equations, we often use algebraic techniques to isolate the variable of interest. For example, we converted the standard form equation \( x - y = 2 \) to slope-intercept form by isolating y:

Subtract x from both sides: \[ -y = -x + 2 \] Multiply both sides by -1 to get y by itself: \[ y = x - 2 \] These algebraic steps help us understand the line's behavior in a simple manner.

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Most popular questions from this chapter

The city mpg, \(x\), and highway mpg, \(y\), of two cars are given by the points (29,40) and (19,28) . Find a linear equation that models the relationship between city mpg and highway mpg.

Find an equation of a line parallel to the given line and contains the given point. Write the equation in slope-intercept form. line \(2 x-y=6,\) point (3,0)

Graph the linear inequality \(y<4\)

Find the equation of each line. Write the equation in slope-intercept form. \(m=-\frac{3}{5},\) containing point (10,-5)

Find the equation of a line containing the given points. Write the equation in slope-intercept form. (-5,-3) and (4,-6)
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